The Gutzwiller Density Functional Theory
IV) Further developments
I) Introduction
II) Gutzwiller variational theory
Jörg Bünemann, BTU Cottbus
III) The Gutzwiller density functional theory
1. Model for an H2-molecule
2. Transition metals and their compounds
1. Gutzwiller wave functions
2. Ferromagnetism in two-band models
3. Magnetic order in LaFeAsO
1. Remainder: The Density-Functional Theory
2. The Gutzwiller Kohn-Sham equations
3. Example: lattice parameters of iron pnictides
[2. The time-dependent Gutzwiller theory]
1. Superconductivity: beyond the Gutzwiller approximation
I) Introduction
1. Model for an H2-molecule:
energies:
: Coulomb interaction (intra-atomic)
(spin: )
t
t
t t
basis:
matrix element for transitions:
t
1.1. Perspective of elementary quantum mechanics:
ground state:
matrix
(ground-state energy)
1.2. Perspective of solid-state theory
ground state:
('molecular orbitals': 'bonding' & 'anti-bonding')
first: solve the 'single-particle problem' ( ) :
ground state is 'single-particle product state'
Effective single-particle ('Hartree-Fock') theory:
Idea: find the single-particle product state with the lowest
energy
Problems:
('correct' spin symmetry: eigenstate of with eigenvalue )
energy
Hartree-Fock ground state breaks
i) for , is never a single-particle product state
ii) the HF ground state with the 'correct' spin symmetry is
iii)
spin-symmetry for :
1.3. Perspective of many-particle theory
(operator for double occupancies)
with increasing , double occupancies are more
and more suppressed in the ground stateGutzwiller's idea:
Gutzwiller wave function: M. C. Gutzwiller PRL 10, 159 (1963) ( )
compare:
proper choice of the variational parameter
reproduces the exact ground state
2. Transition metals and their compounds
partially filled d-shell: with and
2.1. Transition-metal atoms
in cubic environment:orbitals
orbitals
with the local 'atomic' Hamiltonian:
('multi-band Hubbard models”)
combined spin-orbital index
2.2. Lattice models
Hamiltonian for transition metals with partially filled d-shells:
local Coulomb interaction
orbital energies
('single-band Hubbard model')
Local Hamiltonian for d-orbitals in cubic environment:
with 10 independent parameters
in spherical approximation: 3 Racah parameters
use mean valuesalternatively:
and
3d wave functions are rather localised
in solids, the local Coulomb interaction and the band-width are of the same order of magnitude
- magnetism
- metal-insulator transitions
- magneto-resistance ('giant', 'colossal')
- orbital order
- high-temperature superconductivity
Experiment:
Theory: effective single-particle theories often fail
simplest example: fcc nickel
2.3. Interaction effects in transition-metal compounds
(Hartree-Fock or local density approximation in density-functional theory)
1. Gutzwiller wave functions
for the single-band Hubbard model
one defines: (Gutzwiller wave function)
mit i) arbitrary single-particle product wave function
ii)
alternative formulation:
with 'atomic' eigenstates
and variational parameters :
1.1 Definitions
und
II) Gutzwiller variational theory
multi-band Hubbard models
: atomic eigenstates with energies
e.g.: 3d-shell: dimensional local Hilbert space
10 spin-orbitals
with
generalised Gutzwiller wave function:
: matrix of variational parameters (in this lecture )
problem: is still a complicated many-particle wave function
cannot be evaluated in general
mit
1.2. Evaluation: Diagrammatic expansion
We need to calculate
is a single-particle product wave function
Wick theorem applies, e.g.,
Diagrammatic representation of all terms, e.g.,
with lines
1.3. Simplifications in infinite dimensions
A) Diagrams with three or more lines
Kinetic energy per site on a D-dimensional lattice:
for
(only n.n. hopping)
B) Diagrams with two lines
Idea: make sure that all these diagrams cancel each other
This is achieved by the (local) constraints
C) Disconnected diagrams are cancelled by the norm
Conclusion: All diagrams with internal vertices vanish
The remaining evaluation is rather straightforward:
='renormalisation matrix'
with
1.4. Summary: energy functional in infinite spatial dimensions
(effective hopping parameters)
in the limit of infinite dimensions leads to
and are analytic functions of
and
Evaluation of
Remaining numerical problem:
Minimisation of with respect to and
1.5. Minimisation of the energy functional
We consider as a function of and of the non-interacting
density matrix with the elements
'non-interacting'
Minimisation with respect to leads to the effective single
particle equation
bands are i) mixed ( ) and shifted ( )
ii) renormalised ( )
with
2. Ferromagnetism in two-band models
16 local states:
local Hamiltonian:
-orbitals:
and
1 2
orbital
2-particle states energy Sym.
-orbitals on a simple cubic 3-dimensional lattice
density of states:Gutzwiller wave function:
i) no multiplet coupling
ii) spin-polarisedFermi sea
:
Known: ferromagnetism is hardly found in single-band models
requires pathological densities of states and/or
very large Coulomb parameters
orbital degeneracy is an essential ingredient
therefore, we consider:
with hopping to nearest and next-nearest neighbours
Results:
phase diagramMagnetisation ( )
condensation energy
i) Orbital degeneracy and exchangeinteraction are essential forferromagnetic order
ii) Single-particle approaches areinsufficient
size of the local spin
3. Magnetic order in LaFeAsO
3.1 Electronic structure of LaFeAsO
i) Metal with conductivity mainly in FeAs layers
ii) AF ground state with a magnetic moment of
Y. Kamihara et al., J. Am. Chem. Soc 130,3296 (2008)
(K. Ishida et al. J. Phys. Soc. Jpn. 78, 062001 (2009)) ( )
(DFT: )
3.2 LDA band-structure and effective tight-binding models
Eight-band model Five-band model:
Three-band model:
O. K Andersen and L. Boeri, Ann. Physik 523, 8 (2011)( )
(S. Zhou et al., PRL 105, 096401 (2010))
(S. Graser et al., New J. Phys. 11, 025016 (2009))
3.3 Magnetic order in three-band models
Without spin-flip terms:
(S. Zhou et al., PRL 105, 096401 (2010))
With spin-flip terms:
Hartree-Fock: Conclusion (?):
Gutzwiller theory yields a reasonable magnetic moment without fine-tuningof model parameters
3.4 Five-band model
A) Hartree-Fock:
Conclusions:
- small magnetic moments appear only in a small range of correlation parameters
- no orbital order (in contrast to Hartree-Fock)
still no satisfactory explanation for the magnetic orderobserved in LaFeAsO
T. Schickling et al., PRL 106, 156402 (2011) ( ) 3.4 Five-band model
3.5 Electronic properties of LaAsFeO (eight-band model)
Phase diagram: Magnetic moment:
Reasonable values of the magnetic moment over alarge range of Coulomb parameters
T. Schickling et al., PRL 108, 036406 (2012) ( )
III) The Gutzwiller Density Functional Theory
1. Remainder: The Densitiy Functional Theory
Electronic Hamiltonian in solid-state physics
Hohenberg-Kohn theorem:
Existence of a universal functional of the density such that
has its minimum at the exact ground-state density of
('universal' = independent of )
One usually writes
with : 'kinetic energy functional'
: 'exchange correlation functional'
common approximations:free electron gas
free electron gas
HF approximation for
Kohn-Sham scheme:
Instead of , consider the effective single-particle Hamiltonian
with the 'Kohn-Sham potential' Kinetic energy of non-interacting
particles
and have the same ground-state density
Kohn-Sham equations:
We introduce a basis of local orbitals
2. The Gutzwiller Kohn-Sham equations
We distinguish 'localised' ( ) and 'delocalised' orbitals ( )
where and
are now functionals of the density
Problem: Coulomb interaction is counted twice in the localised
orbitals 'double-counting problem'
Density in the Gutzwiller ground state
depends on and
Gutzwiller-DFT energy functional:
Minimisation
leads to 'Gutzwiller Kohn-Sham equations' with
and
for( )
Correlation-induced changes of :
i)
differs from the DFT expression
ii) Correlated bands are shifted (via )
change of change of
iii) Correlated bands are renormalised in
change of and in
change of
Problems:
1. The local Coulomb interactions are usually
considered as adjustable parameters
'ab-initio' character is partially lost
2. Double-counting problem:
Coulomb interaction appears in and in
One possible solution: subtract
from
3. Example: Lattice parameters of iron pnictides
Interlayer distance and (average) band renormalisation
from G. Wang et. al, Phys. Rev. Lett. 104, 047002 (2010)
Elastic constants:
softening of the corresponding phonon mode
in agreement with experiment
1. Superconductivity: beyond the Gutzwiller approximation
1.1 Diagrammatic expansion
In the single-band case we need to calculate
Procedure:
A) Proper choice of the expansion parameter
B) Use of Wick's theorem and the linked-cluster theorem
Diagrammatic representation
C) Numerical evaluation in real space
i)
ii)
iii)
J. Bünemann et al., EPL 98, 27006 (2012) ( )
IV) Further developments
This fixes three parameters and it remains only one ( or )
A) Proper choice of the expansion parameter
Main idea: Choose parameters such that
with!
Main advantage of the HF-operators: no 'Hartree bubbles'
e.g.:
with
In contrast:
(Wick's theorem)
,
The missing of Hartree bubbles has two consequences:
i) Number of diagrams is significantly reduced, e.g.
Diagrams are fairly localised and the power series in converges rapidly (can be tested for )
ii) Each line is ( number of spatial dimensions)
This suggests the following strategy:
i) Calculate
in momentum space (i.e., with negligible numerical error)
ii) Calculate all diagrams (power series in ) in real space up
iii) Minimise the energy with respect to
to a certain order in
1.2 The one-dimensional Hubbard model
In one dimension one can evaluate Gutzwiller wave functions exactly [W. Metzner and D. Vollhardt, PRL 59, 121 (1987)]
convergence of our approach can be tested
With the exact results we may calculate analytically each order of double occupancy and the kinetic energy :
i) Double occupancy ii) Kinetic energy
1.3 Fermi-surface deformations in two dimensions
The Gutzwiller wave function
,
contains as variational objects the parameters (i.e., ) and the wave function . Without breaking translational or spin-Symmetry the only remaining degree of freedom in a one-band model is the shape of the Fermi surface:
Only constraint:
(particle number conservation)
Minimisation with respect to :
with
,
Pomeranchuk instabilities
According to fRG calculations it may happen that the Fermi surface spontaneously breaks the rotational symmetry of the system at finite ('Pomeranchuk instability').
But:
i) fRG is a perturbative approach
ii) Little is know quantitatively, in particular for larger values of .
Obvious question:
Do we find a Pomeranchuk instabilityin our approach?
1.4 Hamiltonian with only nearest-neighbor hopping
'Normal' Fermi surface:
Pomeranchuk phase:
1.5 Superconductivity in two-dimensional Hubbard models
Pairing beyond is relevant
collaboration with J. Kaczmarczyk, Krakow ( )
Consequence 1: enhancement of stability region
Consequence 2: gap-structure
Phase diagram ( )
with
“Kubo formula”:
(Fourier transformation)
2.1 Two-particle excitations: linear response theory
Aim: calculate the density matrix with the elements
with
equation of motion
2. The time-dependent Gutzwiller theory
2.2 Time-dependent Hartree-Fock Theory (RPA):
Equation of motion
Approximation:
decouples (Wick's theorem)
is a closed set of differential equations for
In linear order with respect to this leads to the RPA result
with and
with
is assumed to be a single-particle wave function
2.3 The time-dependent Gutzwiller theory G. Seibold und J. Lorenzana PRL 86, 2605 (2001) ( )
Comparison:
RPA Time-dep. Gutzwiller theory
“Stoner continuum”
Spin-flip-excitation in the “exchange field”
Stoner criterion: ferromagnetism
finite exchange interaction
with
2.4 RPA for the spin susceptibility of a single-band model
2.5 Spin excitations in single-band models
A) Hypercubic lattices in infinite dimensions
i) simple cubic (sc) ii) half cubic (hc)
densities of states
Two-particle response functions in large dimensions only depend on the real parameter
i) Phase diagram of the hc-lattice:
instability of the paramagnet: spin-wave stability
(F. Günther et al., PRL 98, 176404 (2007))
(DMFT: G. Uhrig, PRL 77, 3629 (1996))
instability of the paramagnet:
stability of the ferromagnet:
ii) Phase diagram of an sc lattice:
stablespin waves
(DMFT: Obermeier et al., PRB 56, 8479 (1997))
2.6 Spin excitations in a two-band model
Hartree-Fock phase diagrams:
Gutzwiller phase diagrams:
V) Conclusions
1. The Gutzwiller variational approach provides with us a numerically
'Cheap' way to study multi-band Hubbard models for transition
metals and their compounds.
2. The modest numerical efforts of this method make it particularly
suitable for a self-consistent merger with the DFT.
3. Apart from ground-state properties the method allows us to
calculate quasi-particle excitations and two-particle response
functions.
4. Systematic improvements of the infinite D limit are possible and
sometimes necessary to study, e.g., correlation-induced forms of
superconductivity.